Skip to main content
SpringerLink
Log in

A second-order cone cutting surface method: complexity and application

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p second-order cone cuts in O(plog (p+1)) Newton steps, and that the overall algorithm is polynomial. From the application viewpoint, we implement our algorithm on mixed linear-quadratic-semidefinite programming problems with bounded feasible region and report some computational results on randomly generated fully dense problems. We compare our CPU time with that of SDPLR, SDPT3, and SeDuMi and show that our algorithm outperforms these software packages on problems with fully dense coefficient matrices. We also show the performance of our algorithm on semidefinite relaxations of the maxcut and Lovasz theta problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Atkinson, D.S., Vaidya, P.M.: A cutting plane algorithm that uses analytic centers. Math. Program. Ser. B 69, 1–43 (1995)

    MathSciNet  Google Scholar 

  3. Basescu, V.L., Mitchell, J.E.: An analytic center cutting plane approach for conic programming. Rensselaer Polytechnic Institute, Troy, NY 12180 (June 2005)

  4. Borchers, B.: SDPLIB 1.2, a library of semidefinite programming problems. Optim. Methods Softw. 11, 613–623 (1999)

    Article  MathSciNet  Google Scholar 

  5. Burer, S., Choi, C.: Computational enhancements in low-rank semidefinite programming. Optim. Methods Softw. 21(3), 493–512 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. Ser. B 95(2), 329–357 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burer, S., Monteiro, R.D.C.: Local minima and convergence in low-rank semidefinite programming. Math. Program. Ser. A 103(3), 427–444 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chua, S.K., Toh, K.C., Zhao, G.Y.: An analytic center cutting plane method with deep cuts for semidefinite feasibility problems. J. Optim. Theory Appl. 123, 291–318 (2004)

    Article  MathSciNet  Google Scholar 

  9. Goffin, J.-L., Vial, J.-P.: Multiple cuts in the analytic center cutting plane methods. SIAM J. Optim. 11, 266–288 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Goffin, J.-L., Luo, Z.-Q., Ye, Y.: Complexity analysis of an interior cutting plane for convex feasibility problems. SIAM J. Optim. 6, 638–652 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Helmberg, C.: Semidefinite programming for combinatorial optimization. Habilitationsschrift, ZIB-Report 00-34 (October 2000)

  12. Helmberg, C.: Numerical evaluation of SB method. Math. Program. 95, 381–406 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kim, S., Kojima, M., Yamashita, M.: Second order cone programming relaxation of a positive semidefinite constraint. Optim. Methods Softw. 18(5), 535–541 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Krishnan, K., Mitchell, J.E.: A unifying framework for several cutting plane methods for semidefinite programming. Optim. Methods Softw. 21(1), 57–74 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Krishnan, K., Mitchell, J.E.: A semidefinite programming based polyhedral cut-and-price approach for the maxcut problem. Comput. Optim. Appl. 33(1), 51–71 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lewis, A.S., Overton, M.L.: Eigenvalue optimization, Acta Numer., 149–190 (1996)

  18. Luo, Z.-Q., Sun, J.: An analytic center based on column generation algorithm for convex quadratic feasibility problems. SIAM J. Optim. 9(1), 217–235 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Luo, Z.-Q., Sun, J.: A polynomial cutting surfaces algorithm for the convex feasibility problem defined by self-concordant inequalities. Comput. Optim. Appl. 15, 167–191 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lüthi, H.J., Büeler, B.: The analytic center quadratic cut method (ACQCM) for strongly monotone variational inequality problems. SIAM J. Optim. 10(2), 415–426 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mitchell, J.E., Basescu, V.L.: Selective Gram-Schmidt orthonormalization for conic cutting surface algorithms. Technical report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 (December 2006). Accepted for publication in Mathematical Methods of Operations Research

  22. Mitchell, J.E., Todd, M.J.: Solving combinatorial optimization problems using Karmarkar’s algorithm. Math. Program. 56, 245–284 (1992)

    Article  MathSciNet  Google Scholar 

  23. Oskoorouchi, M.R., Goffin, J.L.: The analytic center cutting plane method with semidefinite cuts. SIAM J. Optim. 13(4), 1029–1053 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Oskoorouchi, M.R., Goffin, J.L.: An interior point cutting plane method for the convex feasibility problem with second-order cone inequalities. Math. Oper. Res. 30(1), 127–149 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Oskoorouchi, M.R., Goffin, J.L.: A matrix generation approach for eigenvalue optimization. Math. Program. Ser. A 109, 155–179 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Overton, M.L.: Large-scale optimization of eigenvalues. SIAM J. Optim. 2, 88–120 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sharifi Mokhtarian, F., Goffin, J.-L.: An analytic center quadratic cut method for the convex quadratic feasibility problem. Math. Program. Ser. A 93(2), 305–325 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sivaramakrishnan, K.: A parallel conic interior point decomposition approach for BLOCK-ANGULAR semidefinite programs. Technical Report, Department of Mathematics, North Carolina State University, Raleigh, NC, 27695 (December 2006)

  29. Sivaramakrishnan, K., Plaza, G., Terlaky, T.: A conic interior point decomposition approach for large scale semidefinite programming. Technical Report, Department of Mathematics, North Carolina State University, Raleigh, NC, 27695 (December 2005)

  30. Sonnevend, G.: New algorithms in convex programming based on a notation of center and on rational extrapolations. Int. Ser. Numer. Math. 84, 311–327 (1988)

    MathSciNet  Google Scholar 

  31. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  32. Sturm, J.F.: Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optim. Methods Softw. 17(6), 1105–1154 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sun, J., Toh, K.C., Zhao, G.Y.: An analytic center cutting plane method for semidefinite feasibility problems. Math. Oper. Res. 27(2), 332–346 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3—a Matlab software package for semidefinite programming, version 2.1. Optim. Methods Softw. 11, 545–581 (1999)

    Article  MathSciNet  Google Scholar 

  35. Toh, K.C., Zhao, G.Y., Sun, J.: A multiple-cut analytic center cutting plane method for semidefinite feasibility problems. SIAM J. Optim. 12(4), 1126–1146 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Tutuncu, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95, 189–217 (2003)

    Article  MathSciNet  Google Scholar 

  37. Ye, Y.: A potential reduction algorithm allowing column generation. SIAM J. Optim. 2, 7–20 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ye, Y.: Complexity analysis of the analytic center cutting plane method that uses multiple cuts. Math. Program. 78, 85–104 (1997)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Business Administration, California State University San Marcos, San Marcos, CA, 92096, USA

    Mohammad R. Oskoorouchi

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    John E. Mitchell

Authors
  1. Mohammad R. Oskoorouchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John E. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad R. Oskoorouchi.

Additional information

M.R. Oskoorouchi’s work has been completed with the support of the partial research grant from the College of Business Administration, California State University San Marcos, and the University Professional Development Grant.

J.E. Mitchell’s material is based upon work supported by the National Science Foundation under Grant No. 0317323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oskoorouchi, M.R., Mitchell, J.E. A second-order cone cutting surface method: complexity and application. Comput Optim Appl 43, 379–409 (2009). https://doi.org/10.1007/s10589-007-9141-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9141-x

Keywords

Search

Navigation